Swenson and Faulkner, and Boothroyd et al. investigated the possibility that early main-sequence mass loss via a stronger early solar wind could be responsible for the observed solar lithium and beryllium depiction. This depletion requires a total mass loss of {approximately}0.1 M{circle_dot}, nearly independent of the mass loss timescale. We have calculated the evolution and oscillation frequencies of solar models including helium and element diffusion, and such early solar mass loss. We show that extreme mass loss of 1 M{circle_dot} is easily ruled out by the low-degree p-modes that probe the solar center and sense the steeper molecular weight gradient produced by the early phase of more rapid hydrogen burning. The effects on central structure are much smaller for models with an initial mass of 1.1 M{circle_dot} and exponentially-decreasing mass loss irate with e-folding timescale 0.45 Gyr. While such mass loss slightly worsens the agreement between observed and calculated low-degree modes, the observational uncertainties of several tenths of a microhertz weaken this conclusion. Surprisingly, the intermediate-degree modes with much smaller observational uncertainties that probe the convection zone bottom prove to be the key to discriminating between models: The early mass loss phase decreases the total amount of helium and heaviermore » elements diffused from the convection zone, and the extent of the diffusion produced composition gradient just below the convection zone, deteriorating the agreement with observed frequencies for these modes. Thus it appears that oscillations can also rule out this smaller amount of gradual early main-sequence mass loss in the young Sun. The mass loss phase must be confined to substantially under a billion years, probably 0.5 Gyr or less, to simultaneously solve the solar Li/Be problem and avoid discrepancies with solar oscillation frequencies.« less